Integrand size = 10, antiderivative size = 26 \[ \int e^x \sinh ^2(4 x) \, dx=-\frac {1}{28} e^{-7 x}-\frac {e^x}{2}+\frac {e^{9 x}}{36} \]
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Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2320, 12, 276} \[ \int e^x \sinh ^2(4 x) \, dx=-\frac {1}{28} e^{-7 x}-\frac {e^x}{2}+\frac {e^{9 x}}{36} \]
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Rule 12
Rule 276
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (1-x^8\right )^2}{4 x^8} \, dx,x,e^x\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \frac {\left (1-x^8\right )^2}{x^8} \, dx,x,e^x\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \left (-2+\frac {1}{x^8}+x^8\right ) \, dx,x,e^x\right ) \\ & = -\frac {1}{28} e^{-7 x}-\frac {e^x}{2}+\frac {e^{9 x}}{36} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int e^x \sinh ^2(4 x) \, dx=-\frac {1}{28} e^{-7 x}-\frac {e^x}{2}+\frac {e^{9 x}}{36} \]
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Time = 0.64 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65
method | result | size |
parallelrisch | \(-\frac {{\mathrm e}^{x} \left (\cosh \left (8 x \right )+63-8 \sinh \left (8 x \right )\right )}{126}\) | \(17\) |
risch | \(\frac {{\mathrm e}^{9 x}}{36}-\frac {{\mathrm e}^{x}}{2}-\frac {{\mathrm e}^{-7 x}}{28}\) | \(18\) |
default | \(-\frac {\sinh \left (x \right )}{2}+\frac {\sinh \left (7 x \right )}{28}+\frac {\sinh \left (9 x \right )}{36}-\frac {\cosh \left (x \right )}{2}-\frac {\cosh \left (7 x \right )}{28}+\frac {\cosh \left (9 x \right )}{36}\) | \(34\) |
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (17) = 34\).
Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.35 \[ \int e^x \sinh ^2(4 x) \, dx=-\frac {\cosh \left (x\right )^{8} - 64 \, \cosh \left (x\right )^{7} \sinh \left (x\right ) + 28 \, \cosh \left (x\right )^{6} \sinh \left (x\right )^{2} - 448 \, \cosh \left (x\right )^{5} \sinh \left (x\right )^{3} + 70 \, \cosh \left (x\right )^{4} \sinh \left (x\right )^{4} - 448 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{5} + 28 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{6} - 64 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 63}{126 \, {\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).
Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int e^x \sinh ^2(4 x) \, dx=\frac {31 e^{x} \sinh ^{2}{\left (4 x \right )}}{63} + \frac {8 e^{x} \sinh {\left (4 x \right )} \cosh {\left (4 x \right )}}{63} - \frac {32 e^{x} \cosh ^{2}{\left (4 x \right )}}{63} \]
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Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int e^x \sinh ^2(4 x) \, dx=\frac {1}{36} \, e^{\left (9 \, x\right )} - \frac {1}{28} \, e^{\left (-7 \, x\right )} - \frac {1}{2} \, e^{x} \]
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Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int e^x \sinh ^2(4 x) \, dx=\frac {1}{36} \, e^{\left (9 \, x\right )} - \frac {1}{28} \, e^{\left (-7 \, x\right )} - \frac {1}{2} \, e^{x} \]
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Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int e^x \sinh ^2(4 x) \, dx=\frac {{\mathrm {e}}^{9\,x}}{36}-\frac {{\mathrm {e}}^{-7\,x}}{28}-\frac {{\mathrm {e}}^x}{2} \]
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